[of Corollary REF ]
Corollary REF  follows from recursion once we show that the \(Q\) th stage random survival forest is consistent and \(n^{(Q)}> cn\)  for some constant \(c\)  almost surely.
For each \(h\in \mathcal {H}^{(Q)}\)  and treatment \(a \in \mathcal {A}^{(Q)}\) , Assumption REF  guarantees that the probability of a patient making all planned visits is at least \(M^{Q-1}\) . That is, \(n^{(Q)}\ge M^{(Q)}n\)  almost surely.
Uniform consistency of the stage-wise random survival forest again relies on arbitrarily small node size, \(\sup _{h\in \mathcal {H}} \sup _{h^{\prime } \in k_h} \Vert h - h^{\prime }\Vert _1\) . Although the support in Assumption REF  is different from that in Assumption REF , the bounded marginal density in Assumption REF  is sufficient to follow the same lines of proof given in [1]} and [2]}.
Now, by having \(S^{\prime }(t\mid h) = \tilde{S}^{\prime }(t\mid h) = 1(t \le 0)\)  for all \(h\) , Theorem REF  yields uniform consistency of \(\hat{S}^{(Q)}(\cdot \mid \cdot )\) . Consistency of the rest of the sequence then follows as \(n^{(q)}\ge n^{(Q)}\)  for \(q = Q-1, Q-2, ..., 1\) .
